How many fractions lie between 1/3 and 1/2 in a sorted set of reduced proper fractions?

Solution:

If you haven't done so already, read about Farey sequences in Wikipedia
http://en.wikipedia.org/wiki/Farey_sequence, where you will learn about
mediants. Then divide and conquer. The number of Farey ratios between
(a, b) is 1 + the number between (a, mediant a b) + the number between
(mediant a b, b). Henrylaxen 2008-03-04